Structural Flexibility and Mounts

7/25/2019 Structural Flexibility and Mounts

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NASA Technical Memorandum 85725

NASA-TM-85725 19840008522

Effect of Structur

a

l

F

lexibility

o

n the Design of Vibr

a

tion-Isol

a

ting

M

ounts for Aircr

a

ft Engines

Willi

a

m H. Phillips

FEBRUARY 1984

,_,_.'

T

:

._t.

_A _A


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3/28

NASA Technical Memorandum 85725

Effect of Structur

a

l

F

lexibility

on the Design of Vibr

a

tion-Isol

a

ting

M

ounts for Aircr

a

ft Engines

Willi

a

m H. Phillips

La

n

gley Research Center

Hampton Virginia

N

I

LSA

National Aeronautics

and Space Administration

ScientificandTechnical

Informa

t

ion O

f

fice

19

8

4


4/28

I

I


5/28

SUMMARY

In a pair of reports published in 1938 and 1939, E. S. Taylor and K. A. Browne

point out the advantages of designing vibration-isolating engine mounts to decouple

the rotational and linear oscillations of the power plant and present a technique for

providing decoupling when the mounting points are located behind the center of grav-

ity of the power plant. _nese authors assume that the mount structure is rigid. _ne

present analysis extends this analysis to include the effects of structural flexibil-

ity. %_e results of this analysis show that the effects of structural flexibility

increase as the distance between the center of gravity of the power plant and the

plane of the mounting pads increases. Equations are presented to allow the design of

mount systems, and data are presented to illustrate the results for a range of design

conditions.

INTRODUCTION

In a pair of reports published in 1938 and 1939 (refs. I and 2), E. S. Taylor

and K. A. Browne discuss the problem of vibration isolation of aircraft power plants

and present a technique for decoupling linear and rotational oscillations of a power

plant when it is attached at mounting points located behind the center of gravity of

the engine-propeller combination. An example of decoupling is a condition such that

a vertical force applied at the center of gravity of the power plant produces only

vertical motion and no pitching, and a pitching moment applied to the power plant

produces only pitching and no vertical movement of the center of gravity. The

decoupled condition is very desirable because it allows separate consideration of the

static deflections and the natural frequencies involving translation and rotation,

and because it decreases the response of many resonant modes to linear or rotational

excitations. This mounting technique has been widely used on aircraft radial engines

and on horizontally opposed engines used in general aviation airplanes.

The mounting technique described in references I and 2 consists of using rubber

mounts with different stiffnesses along three mutually perpendicular axes. An equa-

tion is derived in reference 1 for the angle at which these mounts should be placed

to obtain the desired decoupling of linear and rotational motions. This technique of

arranging the mounts so that their axes intersect at a point on a principal axis of

the body to be isolated has been given the name "focal isolation system" and has been

used in applications other than aircraft engines. A general discussion of this tech-

nique is given in reference 3.

In the derivation of reference I, the airplane structure to which the mounts are

attached is assumed to be infinitely stiff. In practice, the structure has some

flexibility. In the present report, the analysis is extended to include the effects

of structural flexibility. In addition, the analysis gives some insight into the

sensitivity of the decoupling to the parameters of the mount design.

Thorough discussions of excitation forces, power-plant vibration modes, and

practical engine-mount design considerations are given in references 1 and 2. The

present report is concerned solely with the effect of structural flexibility on

decoupling. The reader should refer to the reference reports for a more general

discussion of the engine-mount design problem.


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The overall problem of vibration isolation of an aircraft power plant is a prob-

lem in dynamics which can be solved exactly only by accounting for the distribution

of mass and stiffness throughoutthe structure as well as such factors as engine and

propeller gyroscopic effects. The present analysis, however, treats a problem in

statics. The results of this simple approach are useful in giving equations for the

design of the vibration-isolating mounts. These equations are believed to be

applicable in the usual situation in which the engine-mount structure itself is

relatively light and is placed between the large mass of the engine and other heavy

components of the airplane.

ANALYSIS

The analysis given herein follows the notation of reference I (where possible)

to facilitate a comparison of the results. For completeness, the derivation is given

starting frgm basic principles. The analysis, therefore, repeats some of the steps

given in reference I.

Consider the engine-mount system shown in figure I. The engine is supported

by m equally spaced mounting pads, and each pad has its axis intersecting the cen-

terline of the engine at a common point and making an angle e with the centerline.

(A list of symbols used in this paper appears after the references.) Each pad has

spring rates kA in the tangential direction of the mounting circle, kB in the

axial direction of the pad, and kC in the radial direction normal to the axis of

the pad. The positive directions of the mounting pad axes are shown in the figure.

For analysis, the vertical and pitching motions of a power plant are considered.

The same analysis is applicable to lateral decoupling of engine vibrations (latera

movement and yawing rotation) by redefining axes. A vertical force applied at the

center of gravity of the power plant causes deflections of each of the mounting pads

and a vertical deflection and pitching rotation of the airplane structure as shown in

figure 2. If the structural member or ring to which the mounting pads are attached

is assumed to be rigid, this ring will undergo a translation Z and a rotation @.

Positive directions of these quantities are shown in figure 2. In the subsequent

derivation, however, the force Fz is taken as the force applied by the mounting

pads to the engine. This force has a direction opposite from the force applied to

the mount. The rotation of the mount, therefore, is assumed to be given by the

equation

F a

Z

e=

C

e

or (1)

F aZ

C -

e e

where a is the horizontal distance from the engine center of gravity to the mount-

ing pad plane.

This definition of C8 is arbitrary. The actual relation between the force

Fz and the resulting deflection 8 depends on how the structure of the aircraft is

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restrained. The deflection e results from both the moment of the force Fz about

the mount ring location and the force FZ itself. An experimental determination of

C8 would, therefore, give a value dependent on the distance a as well as on the

method of restraint of the structure. In practice, engineering judgment would be

required to select a reasonable value of C@. For this analysis, however, the pre-

ceding formula is adequate, provided the value of C@ is applied to conditions simi-

lar to those under which it was measured or calculated.

The following procedure is used to derive a relation between the mounting pad

angle 5, the ratio a/r (called the overhang ratio), and the stiffness parameters

of the system to produce the desired decoupling. The rear face of the power plant

and the mounting ring itself are assumed rigid. A parallel vertical displacement of

the engine is assumed to occur, as required for decoupling, which produces equal

vertical displacements of the front faces of each mounting pad. The total vertical

force corresponding to this displacement causes a rotation of the mounting ring which

produces a horizontal displacement of the rear face of each mounting pad proportional

to i_s distance above or below the centerline. The accompanying vertical displace-

ment of the mounting ring has no effect on decoupling and can be neglected in the

analysis. The lack of effect of the vertical displacement of the mounting ring

results from the fact that it causes no additional relative displacement of the front

and rear faces of the mounting pads because the power plant moves only vertically and

is free to move the same amount as the mounting ring if the decoupling condition is

satisfied.

From the relative displacements of the front and rear faces of each mounting

pad, the axial, radial, and tangential forces produced on the engine by each pad are

calculated. The forces are resolved along the power-plant axes, allowing the total

vertical force and the moment about the center of gravity of the power plant to be

calculated. The vertical force is set equal to the force used previously in calcu-

lating the rotation of the mounting ring, and the moment is set equal to zero as

required by the assumption of a parallel vertical displacement of the engine. From

the resulting equations, the stiffness components of the mount may be determined and

a relation may be obtained between the mounting pad angle 5, the overhang ratio

a/r, and the stiffness parameters of the system.

The tangential, axial, and radial components of deflection of the front face of

each pad caused by a displacement AZ of the engine are

A = -AZ sin 6

tan

A = -AZ cos 6 sin _ (2)

ax

A = Az cos 6 cos

rad


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The tangential, axial, and radial components of deflection of the rear face of each

mounting pad caused by a rotation 8 of the mounting ring are

1

Atan 0

A = -@r cos 6 cos e (3)x

Arad -@r cos 6 sin

The components of force produced by each mounting pad result from the components of

relative displacement of the front and rear faces. These relative displacements

multiplied by the corresponding spring rates give the components of force produced by

the pad. Tnese components of force are

F = k AZ sin 6

tan A

F = k AZ cos 6 sin _ - k 8r cos 6 cos _ (4)

ax B B

F = -k AZ cos 6 cos _ - k @r cos 6 sin

rad C C

These force components may be resolved along the X-, Y-, and Z-axes as follows:

F = F cos _ + F sin _

X ax rad

Fy F cos 6 - F sin _ sin 6 + F cos _ sin 6 (5)

an ax rad

F = -F sin 6 - F sin _ cos 6 + F cos _ cos 6

Z tan ax rad


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ubstituting equations (4) into equations (5) gives the following equations for the

, Y, and Z components of force produced by each mounting pad:

2

F = k AZ cos 6 sin _ cos e - k 8r cos 6 cos

X B B

2

- kc AZ cos 6 sin e cos e - kc0r cos 6 sin

2

Fy = kA AZ sin 6 cos 6 - kB AZ sin 6 cos 6 sin

2

+ kB0r sin 6 cos 6 sin e cos _ - kc AZ sin 6 cos 6 cos _ (6)

- k 8r sin 6 cos 6 sin e cos

C

FZ -kA AZ sin26 kB AZ cos26 sin2 26

- _ + kB@r cos sin _ cos

2

- kc AZ cos26 cos e - kcSr cos26 sin e cos

e values AZ and @ are the same for all mounting pads. The sum of the forces

or m mounting pads equally spaced in the mounting ring (m > 2) may be shown to be

Fx= 0

7 Fy = 0 (7)

m m

_ FZ 2 _Z (kA + kB sin 2

- -- e + kc cos2e) - _ 8r(kC - kB) sin e cos

nese results depend on the relations

m-1 m-1

E sin 6

=

E cs 6 =m m

i=0 i=0

m-1

m

OS 6m 2

i=0

e values 6m, where 6

=

6 + i --2_ for i

=

0, 1, ..., m - I and m > 2, are the

o .m

gles of m equally spaced mountlng pads around the mounting ring. A proof of

hese relations is given in the appendix.

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If the value of 8 from equation (1) is substituted in the expression for

_ F in equations (7), an equation relating the vertical force to the vertical

Z

displacement of the engine is obtained. (The values of FZ and Fx henceforth are

the total values, and the summation signs are omitted.)

m 2 2 m Fzar

FZ = - _ AZ (kA + kB sin _ + kC cos _) + 2 C@ (kc - kB) sin _ cos _ (8)

or

m 2 2

AZ (kA + kB sin e + kC cos e)

Fz = - (9)ar

I 2 cs(kc - kB) sin e cos

The pitching moment exerted by a mounting pad about the center of gravity of the

engine is

F r cos 6 + F a

X Z

For decoupling, this total pitching moment must be zero,

F r cos 6 + F a = 0

X Z

or

Fx cos 6

a/r = F (I0)

Z

The total value of Fx cos 6 for m equally spaced mounting pads is

m m 2

Fx cos 6 =_ AZ (kB -k C ) sin _ cos _ -_ @r(kC sin _ + kB cos2_) (11)

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bstituting equation (II) into equation (10) and using the value of @ in terms

Fz from equation (I) gives the following expression for a

/

r:

Fzar

2 2

m AZ (kB - kc) sin e cos _ (kC sin e + k cos e)

2 2 C8 B

0a

/

r = - FZ (12)

bstituting the value for Fz from equation (9) into equation (12) gives

m2 AZ (kB - kC) sin _ cos _ [1 m2_@ar(k- k B) sin _ cos _I

a/r = 2

m AZ kA + kB sin _ + kC cos

mar 2 2

c@(kc sin _ + kB cos _) (13)

mplifying gives

kI_c _ 11 m ar 11 k_l _I

C - sin e cos e 2 C0 kc - sin _ cos

a/r = (k kB 2 21

c + k_csin _ + cos

(s c

ar kc in2_ + cos2 (I4)

2C 8

uation (14) gives a relation between the overhang ratio a/r, the mounting pad

gle _, the stiffness parameters kA, kB, and kc, and the nondimensional ratio

m ar

fs - 2 C 8 kc (15)

his quantity is the structural flexibility parameter fs and is related to the

elative stiffness of the mount system and the engine-mount structure.


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As shown in reference I, the values kA, kB, and kC may be given arbitrary

values by combining a system of linkages with rubber mounting units. In many prac-

tical cases, however, particularly for smaller engines, the mounting pads are used

without the linkages. In these cases the tangential and radial stiffnesses kA

and kC are equal, whereas the axial stiffness is larger by a factor f. In prac-

tice, f may be of the order of 10 to 50. Under these conditions, equations (12)

and (14) may be placed in a simpler form. Let

kA = kc = k 1

(16)

k B = fk C = fk

Substituting equations (16) and fs into equation (9) gives, for the vertical mount

stiffness,

_ 2

m k[(f - I) sin e + 2]

2

Fz/AZ = - I + f (f - I) sin _ cos _ (17)

s

Substituting equations (16) and fs into equation (14) gives, for the value of a/r,

f-

(f - 1) sin _ cos _ll

u

+ fs (f - I) sin _ cos J51

a/r = 2 - fs[(f - I) cos2_ + I] (18)

(f - I) sin _ + 2

Another quantity of interest from the standpoint of vibration isolation is the

torsional stiffness of the mounting system for rotation of the power plant about the

longitudinal axis. This quantity is given by the equation

TX _

C_- _ mr2k A (19)

This stiffness is independent of the angle _ of the mounting pads and may be used

to establish the tangential stiffness kA independently of decoupling

considerations.

As pointed out previously, the preceding analysis may also be applied to

decoupling lateral motion and yawing rotation by redefining the axis system.


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RESULTS

The relationbetweenthe overhangratio a

/

r, the mountingpad angle _, and the

stiffnessparameters kA, kB, and kC (eq. (I4)) may be comparedwith the following

equation (eq.(II) of ref. I) for the caseof an infinitelystiff mountingstructure:

I sin 2_

a/r - 2 k + k (20)

A C + sin2

k -k

B C

By algebraicrearrangementof terms,thisequationmay be put in the form

kC - sin _ cos

a/r = /kA kB h (21)

kc_C + _C sin2e + cs2

This equation is in agreement with equation (14) if the structural stiffness C@

goes to infinity. The effect of the flexible structure is contained in two addi-

tional terms involving the flexibility parameter

mar

f - kc (22)

s 2C 8

Tne relation between the mounting pad angle e and the stiffness ratio f for

various values of the overhang ratio a/r and the structural flexibility parameter

fs' as obtained from equation (18), are presented in figure 3. This figure shows the

relation between the parameters required to satisfy the decoupling condition. In

ddition, the variation of vertical mount stiffness with mounting pad angle when the

ecoupling condition is satisfied is shown in figure 4, which also includes replots

f some of the curves of figure 3. The vertical mount stiffness is given as the

F

z

m

_k AZ

In order to determine the value of the stiffness ratio and the corresponding value of

the verticalmount stiffness from figure 4, these values should be read on the

abscissa for a given value of _ on the ordinate. The stiffness presented is the

ratio of force to the vertical deflection of the center of gravity of the power plant

ith respect to the center of the mount ring. Any additional compliance caused by

vertical deflection of the airplane structure should be added to the deflection

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contributed by the mounting pads in order to calculate the overall stiffness of the

mount. As noted previously, the vertical stiffness of the structure does not affect

the decoupling calculations. Since no value was assigned to this stiffness, its

effect cannot be included in the results plotted in figure 4.

Data are not presented for the case of unequal mounting pad stiffness param-

eters kA and kB, but similar results could be plotted for any desired values of

these parameters by using equations (9) and (14).

The results presented in figures 3 and 4 are for the case of mounting pads

equally spaced around the mount ring. Other arrangements of mounting pads may be

studied by using equations (6) and (10) and adding the contribution of the individual

mounting pads. The resulting equations for vertical mount stiffness and for the

decoupling relation are found to have the same form as equations (17) and (18) but

with different values for some of the constants.

For example, for the case of four mounting pads oriented 30 from the horizon-

tal (as shown in the accompanying sketch), the equations for Fz

/

AZ and a

/

r cor-

responding to equations (17) and (18) are

Z

-k[(f - 1) sin2_ + 4]

Fz

/A

Z = f (23)

I + _(f - 1) sin _ cos

(f - 1) sin _ cos _ + (f - 1) sin _ cos f 2

a/r = 2 - _[ (f - 1) cos _ + I] (24)

(f - I) sin e + 4

I0


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2arkC

here fs is C for the case of four mounting pads. These equations are for an

8

rrangement of mounting pads typical of that used for the in-line, horizontally

pposed engines often used in general aviation airplanes.

DISCUSSION

The results given in figure 3 show that for a given stiffness ratio f there

re two values of the mounting pad angle e which will provide decoupling, provided

he value of f is greater than some minimum value. For conditions of small over-

ang ratio, the value of f for decoupling is almost constant over a wide range of

ounting pad angles. The decoupling property is, therefore, quite insensitive to

ounting pad angle provided the stiffness ratio has the correct value. At larger

alues of overhang ratio, the range of suitable mounting pad angles is reduced.

The effects of structural flexibility, as shown by the curves for various values

f the structural flexibility parameter fs are small for small values of overhang

atio but increase rapidly with larger values. The required value of f increases

apidly with both overhang ratio and structural flexibility. Numerical values of the

tructural flexibility parameter may be interpreted by noting that with the pads

riented at _

=

90 and an overhang ratio of 1.0, the angle through which the

ngine would pitch with respect to the mount ring as a result of a vertical force at

ts center of gravity would equal the angular deflection of the mount ring for

s = 1.00. Normally, the stiffness of the mounting pads in their more flexible

irection is expected to be much less than that of the mount structure. Therefore,

alues of fs plotted are limited to 0.05. Furthermore, with large overhang ratios,

ount flexibility prevents decoupling from being attained with practical values Of

if the value of fs is greater than about 0.05.

Though a wide range of mounting pad angles may be selected to provide decou-

ling, the data of figure 4 show that the vertical mount stiffness increases rapidly

ith increasing values of _. The designer, therefore, has the ability to select the

ertical mount stiffness to provide the most effective vibration isolation and still

eet the decoupling condition. As pointed out in reference I, even greater flexibil-

ty in design is provided b

y

mount systems in which the tangential and radial spring

ates kA and kC are different.

A question may arise as to the method of calculating or measuring the mount

tiffness parameter C . If the stiffness is determined in a static test, this value

aries depending on wh_t point in the structure is considered fixed. A more rational

ay to determine C would be to measure or calculate the response of the structure

o a vibratory forc_ applied at the location of the engine center of gravity in the

requency range of the engine vibrations. In this way the restraint on the struc-

ure, which arises mainly from the inertia of various structures and equipment aft of

he mount structure, would be properly represented.

CONCLUDING REMARKS

In previous reports, a technique has been presented for the design of vibration-

solating mounts for a rear-mounted engine to decouple linear and rotational oscilla-

ions of the engine. This technique has been _idely used. The present analysis

xtends this design procedure to account for the flexibility of the structure to

hich the mounts are attached.

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Equations and curves are presented to allow the design of mount systems and to

illustrate the results for a range of design conditions. The results of this analy-

sis show that the structural flexibility has a relatively small effect when the cen-

ter of gravity of the engine is close to the plane of the mounting points, but it

becomes more important as the distance between the center of gravity and the plane of

the mounting points increases.

Langley Research Center

National Aeronautics and Space Administration

Hampton, VA 23665

December 15, 1983

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APPENDIX

PROOF OF RELATIONS INVOLVING ANGULAR POSITIONS OF MOUNTING PADS

This appendix contains proof of a theorem concerning the sum of sines and

cosines of angles which divide a circle into m equal parts (m > 2). The derivation

of equations (7) of the main text depends on the relations

m-1 m-1

_ sin 6 =_ cos 6

=

m m (A1)

i=0 i=0

m-1 m-1

_ sin26m _ cos26 mm = _ (a2)

i=0 i=0

where

2%

6 = 6 + i -- (i = , I, .... m - 1 for m > 2)

m o m

Considerfirst

exp 6o + i _- + exp j 6o + i

os 6 = cos + i =

m o 2

i=0 i=0 i=0

m-1

= _ exp(j6)2exp( 2 _ exp(-j6ji--% + 2 o exp(- 2ji--_ (A3)

i=0

j = _. The terms exp_jil 2_, when the values of i are substituted,here - . m

constitute the terms of a geometrlc series. The s

u

m of the terms of the series

I + r + r2 + ... + rm-1 is given by the f

o

rmula (1 - rm)

/

(1 - r). The summation

equation (A3) may, therefore, be written

exp(j6)I - exp(j2Z)_ + exp(-J6)I -xp(-_J2-_)-__xpi_) j _ expi_) _ (A4)

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APPENDIX

But exp(j2_) = cos 2_ + j sin 2_ = I and exp(-j2_) = cos 2_ - j sin 2_ = I.

Hence, the numerator of each term goes to zero. The denominator remains finite for

m > I in this case.

The same reasoning applies for

m-1

sin 6m

i=O

the only difference being in the coefficients of the terms which go to zero. Hence,

equation (AI) applies for any m > I.

Consider next

mcosm{o I

6 =m cos2(6o + m-_I _= xp[jI6 + i m2_)l + exp[-J (6 +2

i=0 i=0 i=0

= m-_ expI2j (6o + i m2_)]+ 24 exp[-2j (6o + i m2---_-_)I (A5)

i=0

From the formula for the sum of a geometric series, the summation equation (A5) may

be written

exp exp

But exp(j4_) = cos 4_ + j sin 4% = I and exp(-j4_) = cos 4_ - j sin 4_ = I.

Hence, the numerators of the second and third terms go to zero. %_nedenominators

also go to zero for m = 2 but remain finite for m > 2. Similar reasoning applies

for

m-1

2

in 6

m

i=O

Hence, equation (A2) applies for any m > 2.

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REFERENCES

I. Taylor, E. S.; and Browne, K.A.: Vibration Isolation of Aircraft Power Plants.

J. Aeronaut. Sci., vol. 6, no. 2, Dec. 1938, pp. 43-49.

2. Browne, K.A.: Dynamic Suspension - A Method of Aircraft-Engine Mounting. SAE J.

(Trans.), vol. 44, no. 5, May 1939, pp. 185-192.

3. Crede, Charles E.: Vibration and Shock Isolation. John Wiley Sons, Inc.,

c.1951.

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SYMBOLS

a horizontal distance from engine center of gravity to plane of mounting pads

C8 stiffness of engine mounting ring in pitch direction, -aFx/0

C# torsional stiffness of mounting system, TX/_

Ftan,Fax,Frad components of force in tangential, axial, and radial directions

applied to engine by a mounting pad

Fx,Fy,FZ forces applied to engine by the mounting pads in the X, Y, and Z

directions

f ratio of axial stiffness of mounting pad to tangential or radial stiffness

when kA = kC = k

fs structural flexibility parameter, mar

_--k C

8

k valu

e

of ta

n

g

e

ntial or radial stiff

ne

ss of mou

n

ting pad, k = kA = k C

kA tangential spring rate of mounting pad

kB axial spring rate of mounting pad

kc radial spring rate of mounting pad

m number of mounting pads

r radius to centerline of engine mounting pads

TX torque about longitudinal axis

X,Y,Z translations along longitudinal, lateral, and vertical axes

angle between mounting pad axis and longitudinal axis

AZ vertical displacement of engine relative to mount ring

A ,A ,A displacements of front or rear faces of mounting pad in

an ax rad

tangential, axial, and radial directions

6 angular position of mounting pad

8 pitching rotation of engine-mount ring

rotation of engine about longitudinal axis

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21/28

Top side view

of engine X

Rear view

of engine

f

/

_ _

-

c.g.of power plant

\ r ial

/

.

.:

- Y Radial

_- 6__Tangential

Z --Typical mounting pad

Figure I.- Sketch of engine-mount system and definition of axes used in analysis.

c.g. of power plant_-x f_ -

T---.

FZ1

Ji Engine-mount

structure

Figure 2.- Deflection of engine-mount structure caused by a force at the

center of gravity of the power plant.

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22/28

70

6O

50

4O

t--

to

3O

t-

_- 20

o

Structural flexibility

parameter

f

s

]0

04

05-------

- o1 .o2_-

b

0 20 40 60 80 100 ]20

Stiffness ratio, f

(a) a/r = 0.5.

50

_40

6

_ 3o

t--

=20

_ Structural flexibility

parameter, fs

_=]0

03_.04 .

05-

o Ol '02

o io 4 0 60 80 16o ]_o

Stiffness ratio

f

(b) a/r = 1.0.

Figure 3.- Variation of mounting pad angle _ with stiffness ratio f for

decoupling for various values of structural flexibility parameter fs"

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23/28

(c) a/r = 1.5.

Stiffness ratio, f

(d) a

/

r = 2.0.

Figure 3.- Concluded.

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24/28

70

6O

5O

6

d0

_0 Vertical mount

30

l;

n

20

Stru

c

tural flexibility

]0 parameter, fs

05

0

0 20 40 60 80 100 ]20

Stiffness ratio, f

I_ 2 4

6

8 10

1

2

Ver

t

ical mount stiffness, FZ

m

-zRZ_Z

(a) a/r = 0.5.

50

_3o

5

_ I // Vertical mount

i \

p

__,,

n

e

s,

_ m \ _ Structural flexibili ty

_ 10 \ I _ parame

t

er fs

_,_ .05

0

20 40 60 80 lO0 120

Stif fness rat io

f

0 2 4 6 8 lO 12

F7

Vertical mount stiffness, m

Tk _Z

(b) a/r = 1.0.

Figure 4.- Variation of mounting pad angle _ with stiffness ratio f for

decoupling and corresponding values of vertical mount stiffness for two

values of structural flexibility parameter fs"

2O


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40

6

.30 Stiffness 0

_ ratio

04

,= 20 Vertical mount

=- sti

f

fness

.__E Stru

c

tural flexibility

parameter fs

=]0

o .

0

4

_E

0

o 2'0 4'0 ob 8'o ido i_o

Stiffness ratio f

FZ

Vertical mount stiffness,

m A

7

_..

T k

(c) a

/

r = 1.5.

_n

30

_" s

s /

_,.___ 0

3

20 Stiffne

_= ratio ____,__

Vertical [/ -'-7 Structural flexibility

mount .,_ k,.. para meter fs

_ 10 sti

ff

ness -2 _. __

o zo 40 do 80 i do J_o

Stiffness ratio, f

L

o _ 4 _ _ 1'o f_

FZ

Vertical mount stiffness

m

_k z_z

(d

a

/

r = 2.0.

Figure 4.- Concluded.

21


26/28

1.

ReportNo. 2.

G

o

v

er

nm

ent

Acc

ess

i

o

n

No. 3

.

R

e

ci

p

ie

n

t

sC

at

al

ogNo.

NASA TM-85725

4

. Ti

t

le and Subti

t

le 5

.

R

e

por

t

Da

te

EFFECT OF STRUCTURAL FLEXIBILITY ON THE DESIGN OF February 1984

VIBRATION-ISOLATING MOUNTS FOR AIRCRAFT ENGINES 6. PerformingOrganizationCode

505-34-03-02

7. Author(s) 8. PerformingOrganizationReport No.

William H. Phillips L-15704

10

. W

o

rk

U

nit No.

9.

P

erformingOrganizat

i

onNameand

A

d

d

ress

NASA Langley Research Center 11.Contractr GrantNo.

Hampton, VA 23665

13

. T

ype of Report and P

er

iod Co

ve

red

12

.

SponsoringAgen

c

y

N

ameand

A

ddr

ess

Technical Memorandum

National Aeronautics and Space Administration

Washington, DC 20546

1

4.Spon

s

o

r

in

gAgency

Code

15. S

u

pplemen

t

a

r

yNo

t

es

16. Abstract

Previous analyses of the design of vibration-isolating mounts for a rear-mounted

engine to decouple linear and rotational oscillations are extended to take into

account flexibility of the engine-mount structure. Equations and curves are

presented to allow the design of mount systems and to illustrate the results for a

range of design conditions.

1

7. Key

Word

s (

S

ugg

e

st

edb

y

A

utho

r

(s)) 18.

D

ist

rib

uti

o

n

S

t

a

te

m

ent

Aircraft design Vibration isolator. Unclassified - Unlimited

Aircraft propulsion

Power plant vibration

Vibration mode decoupling

Focal isolation systems Subject Category 39

19

. S

ecu

ri

tyClassif

.

(of this

r

epo

r

t) 20. Security

C

lassif.(

o

f th

i

spage)

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.

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Unclassified Unclassified 22 A02

For sale by

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ield Virginia 221

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Documents

Structural Flexibility and Mounts